REFLECTION OPERATOR AND HYPERGEOMETRY I: SL(2,ℝ) SPIN CHAIN

In this paper we consider an open SL(2,ℝ) spin chain, mainly the simplest case of one particle. Eigenfunctions of the model can be constructed using the so-called reflection operator. We obtain several representations of this operator and show its relation to the hypergeometric function. Besides, we prove orthogonality and completeness of one-particle eigenfunctions and connect them to the index hypergeometric transform. Finally, we briefly state the formula for the eigenfunctions in many-particle case. Bibliography: 17 titles.